1,027 research outputs found
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
Delone sets of finite local complexity in Euclidean space are investigated.
We show that such a set has patch counting and topological entropy 0 if it has
uniform cluster frequencies and is pure point diffractive. We also note that
the patch counting entropy is 0 whenever the repetitivity function satisfies a
certain growth restriction.Comment: 16 pages; revised and slightly expanded versio
Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling: Application to the Rashba model with attractive interaction
The functional renormalization group (RG) in combination with Fermi surface
patching is a well-established method for studying Fermi liquid instabilities
of correlated electron systems. In this article, we further develop this method
and combine it with mean-field theory to approach multiband systems with
spin-orbit coupling, and we apply this to a tight-binding Rashba model with an
attractive, local interaction. The spin dependence of the interaction vertex is
fully implemented in a RG flow without SU(2) symmetry, and its momentum
dependence is approximated in a refined projection scheme. In particular, we
discuss the necessity of including in the RG flow contributions from both bands
of the model, even if they are not intersected by the Fermi level. As the
leading instability of the Rashba model, we find a superconducting phase with a
singlet-type interaction between electrons with opposite momenta. While the gap
function has a singlet spin structure, the order parameter indicates an
unconventional superconducting phase, with the ratio between singlet and
triplet amplitudes being plus or minus one on the Fermi lines of the upper or
lower band, respectively. We expect our combined functional RG and mean-field
approach to be useful for an unbiased theoretical description of the
low-temperature properties of spin-based materials.Comment: consistent with published version in Physical Review B (2016
The topology of fullerenes
Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website
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